High accuracy analysis of finite element methods for optimal control problems and its application

Abstract

In this paper, we briefly review the recent advances of superconvergence analysis and related efficient finite element algorithms for PDE-constrained optimal control problems. The emphasis is on the superconvergence of finite element approximations to optimal control problems governed by elliptic and parabolic equations, the superconvergence of mixed finite element approximations to optimal controls of elliptic and Stokes equations, the recovery type a posteriori error estimates and the extrapolation and defect correction methods for optimal controls. We give a survey on the above topics and perspectives for future work are included.